General graph theoretic approaches are common tools in Statistical Physics [1]}, [2]}, [3]}.
Previously, one of us has discussed the notion of spin ice on a general undirected graph \(\mathcal {G}\) , where a spin configuration may be thought of as a directed graph, considered for its Coulomb phase properties, and shown that charge correlations are computable from graph spectral analysis [4]}.
We use the same approach here, but unlike the previous approximated study, we derive exact results for the degeneracy of the ice manifold. Later in the paper we will also introduce a way to obtain also estimates (which are lower bounds), but here we prefer to keep a general discussion.
